The resonance method for large character sums

TL;DR

This paper introduces a new approach using the resonance method and saddle point analysis to establish lower bounds on large character sums for various ranges of N, advancing understanding in analytic number theory.

Contribution

It provides novel lower bounds for large character sums using the resonance method, applicable across almost all ranges of N, which was not previously achieved.

Findings

Established new lower bounds for character sums

Applied resonance method and saddle point analysis effectively

Extended results to almost all ranges of N

Abstract

We consider the size of large character sums, proving new lower bounds for the quantity $\Delta(N,q) = \sup_{\chi\neq \chi_0 mod q} |\sum_{n < N} \chi(n)|$ for almost all ranges of $N$. The results are proven using the resonance method and saddle point analysis.